^{1}and Karl F. Freed

^{1,2,a)}

### Abstract

Many glass-forming fluids exhibit a remarkable thermodynamic scaling in which dynamic properties, such as the viscosity, the relaxation time, and the diffusion constant, can be described under different thermodynamic conditions in terms of a unique scaling function of the ratio ργ/T, where ρ is the density, T is the temperature, and γ is a material dependent constant. Interest in the scaling is also heightened because the exponent γ enters prominently into considerations of the relative contributions to the dynamics from pressure effects (e.g., activation barriers) vs. volume effects (e.g., free volume). Although this scaling is clearly of great practical use, a molecular understanding of the scaling remains elusive. Providing this molecular understanding would greatly enhance the utility of the empirically observed scaling in assisting the rational design of materials by describing how controllable molecular factors, such as monomer structures, interactions, flexibility, etc., influence the scaling exponent γ and, hence, the dynamics. Given the successes of the generalized entropy theory in elucidating the influence of molecular details on the universal properties of glass-forming polymers, this theory is extended here to investigate the thermodynamic scaling in polymer melts. The predictions of theory are in accord with the appearance of thermodynamic scaling for pressures not in excess of ∼50 MPa. (The failure at higher pressures arises due to inherent limitations of a lattice model.) In line with arguments relating the magnitude of γ to the steepness of the repulsive part of the intermolecular potential, the abrupt, square-well nature of the lattice model interactions lead, as expected, to much larger values of the scaling exponent. Nevertheless, the theory is employed to study how individual molecular parameters affect the scaling exponent in order to extract a molecular understanding of the information content contained in the exponent. The chain rigidity, cohesive energy, chain length, and the side group length are all found to significantly affect the magnitude of the scaling exponent, and the computed trends agree well with available experiments. The variations of γ with these molecular parameters are explained by establishing a correlation between the computed molecular dependence of the scaling exponent and the fragility. Thus, the efficiency of packing the polymers is established as the universal physical mechanism determining both the fragility and the scaling exponent γ.

This work is supported by the (U.S.) Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-SC0008631. We are grateful to Jacek Dudowicz and Jack F. Douglas for several useful discussions.

I. INTRODUCTION

II. THERMODYNAMIC SCALING WITHIN THE GENERALIZED ENTROPYTHEORY

A. Generalized entropytheory

B. Testing thermodynamic scaling

C. Comparison with experiment

III. RESULTS AND DISCUSSION

A. Influence of the bending energy

B. Influence of the cohesive energy

C. Influence of the chain length

D. Influence of the side group length

E. Correlations between the scaling exponent and measures of fragility

IV. SUMMARY

### Key Topics

- Polymers
- 58.0
- Entropy
- 31.0
- Thermodynamic properties
- 26.0
- Glass transitions
- 20.0
- Lattice theory
- 19.0

## Figures

The logarithm of relaxation time log (τ) calculated from the generalized entropy theory as a function of the ratio ϕγ/T with γ = 14.0 for various pressures. The computations are performed for a melt of chains with the structure of poly(propylene) (PP) with z = 6, a cell = 2.7 Å, ε/k B = 200 K, E b /k B = 400 K, and N = 8000. The same values of z, a cell , ε, E b , and N are used in the computations presented in Fig. 2 . τ is given in units of seconds, which is also used in Fig. 4 . Thermodynamic scaling holds well for low pressures, but relaxation times for P ≳ 50 MPa deviate from the master curve.

The logarithm of relaxation time log (τ) calculated from the generalized entropy theory as a function of the ratio ϕγ/T with γ = 14.0 for various pressures. The computations are performed for a melt of chains with the structure of poly(propylene) (PP) with z = 6, a cell = 2.7 Å, ε/k B = 200 K, E b /k B = 400 K, and N = 8000. The same values of z, a cell , ε, E b , and N are used in the computations presented in Fig. 2 . τ is given in units of seconds, which is also used in Fig. 4 . Thermodynamic scaling holds well for low pressures, but relaxation times for P ≳ 50 MPa deviate from the master curve.

The logarithm of glass transition temperature log (T g ) as a function of the logarithm of glass transition specific volume log (V g ). The red solid line is a linear fit to the data of slope 13.47 for P ⩽ 50.7 MPa. A linear relationship between log (T g ) and log (V g ) indicates that the relaxation times obey thermodynamic scaling.

The logarithm of glass transition temperature log (T g ) as a function of the logarithm of glass transition specific volume log (V g ). The red solid line is a linear fit to the data of slope 13.47 for P ⩽ 50.7 MPa. A linear relationship between log (T g ) and log (V g ) indicates that the relaxation times obey thermodynamic scaling.

Specific volumes v as a function of temperature T for various pressures. 54 The symbols are experimental data taken from Ref. 55 for atactic PP with high molecular weight, and the lines are results calculated from the generalized entropy theory for a melt of chains with the PP structure with z = 6, E b /k B = 409 K, and N = 8000. The cell volume parameter a cell and cohesive energy ε are adjusted to decrease with pressure in order to better describe the experimental data. The parameters are summarized in Table I .

Specific volumes v as a function of temperature T for various pressures. 54 The symbols are experimental data taken from Ref. 55 for atactic PP with high molecular weight, and the lines are results calculated from the generalized entropy theory for a melt of chains with the PP structure with z = 6, E b /k B = 409 K, and N = 8000. The cell volume parameter a cell and cohesive energy ε are adjusted to decrease with pressure in order to better describe the experimental data. The parameters are summarized in Table I .

(a) The logarithm of relaxation time log (τ) as a function of the ratio ϕγ/T for various E b . (b) The logarithm of glass transition temperature log (T g ) as a function of the logarithm of glass transition specific volume log (V g ) for various E b . The solid lines in (b) are linear fits. The computations are performed for the PP structure with z = 6, a cell = 2.7 Å, ε/k B = 200 K, and N = 8000. The same values of z, a cell , ε, and N are used in the computations presented in Figs. 5 and 7 .

(a) The logarithm of relaxation time log (τ) as a function of the ratio ϕγ/T for various E b . (b) The logarithm of glass transition temperature log (T g ) as a function of the logarithm of glass transition specific volume log (V g ) for various E b . The solid lines in (b) are linear fits. The computations are performed for the PP structure with z = 6, a cell = 2.7 Å, ε/k B = 200 K, and N = 8000. The same values of z, a cell , ε, and N are used in the computations presented in Figs. 5 and 7 .

Scaling exponents γ, estimated from two independent methods, as a function of bending energy E b . The red squares and blue circles are obtained from relaxation times and Eq. (2) , respectively.

Scaling exponents γ, estimated from two independent methods, as a function of bending energy E b . The red squares and blue circles are obtained from relaxation times and Eq. (2) , respectively.

Scaling exponents γ as a function of side group bending energy for various fixed backbone bending energies . The calculations consider a poly(1-pentene) (PPe) structure with z = 6, a cell = 2.7 Å, ε/k B = 200 K, and N = 8000.

Scaling exponents γ as a function of side group bending energy for various fixed backbone bending energies . The calculations consider a poly(1-pentene) (PPe) structure with z = 6, a cell = 2.7 Å, ε/k B = 200 K, and N = 8000.

(a) Isochoric fragility parameter m V as a function of bending energy E b . (b) The product of isobaric expansion coefficient α P at the glass transition point and glass transition temperature T g as a function of bending energy E b at constant pressure (P = 0.1 MPa). The lines are a guide to the eye.

(a) Isochoric fragility parameter m V as a function of bending energy E b . (b) The product of isobaric expansion coefficient α P at the glass transition point and glass transition temperature T g as a function of bending energy E b at constant pressure (P = 0.1 MPa). The lines are a guide to the eye.

Scaling exponent γ and isochoric fragility parameter m V as a function of cohesive energy ε. The computations are performed for a polymer melt with the PP structure with z = 6, a cell = 2.7 Å, E b /k B = 400 K, and N = 8000.

Scaling exponent γ and isochoric fragility parameter m V as a function of cohesive energy ε. The computations are performed for a polymer melt with the PP structure with z = 6, a cell = 2.7 Å, E b /k B = 400 K, and N = 8000.

Scaling exponent γ and isochoric fragility parameter m V as a function of chain length N. The computations are performed for a polymer melt with the PP structure with z = 6, a cell = 2.7 Å, ε/k B = 200 K, and E b /k B = 400 K. The smallest value for N is 8.

Scaling exponent γ and isochoric fragility parameter m V as a function of chain length N. The computations are performed for a polymer melt with the PP structure with z = 6, a cell = 2.7 Å, ε/k B = 200 K, and E b /k B = 400 K. The smallest value for N is 8.

Scaling exponent γ and isochoric fragility parameter m V as a function of side group length n. The computations are performed for the poly(α-olefin) structure with z = 6, a cell = 2.9 Å, ε/k B = 200 K, K, K, and N = 8000.

Scaling exponent γ and isochoric fragility parameter m V as a function of side group length n. The computations are performed for the poly(α-olefin) structure with z = 6, a cell = 2.9 Å, ε/k B = 200 K, K, K, and N = 8000.

Correlations between isobaric fragility parameter m P at P = 0.1 MPa and isochoric fragility parameter m V when varying individual molecular parameters. The lines are a guide to the eye.

Correlations between isobaric fragility parameter m P at P = 0.1 MPa and isochoric fragility parameter m V when varying individual molecular parameters. The lines are a guide to the eye.

Correlations between scaling exponent γ and isochoric fragility parameter m V when varying individual molecular parameters. The lines are linear fits according to Eq. (6) with the fitting parameters, γ0 = −8.65 and b = 847 for chains with the poly(propylene) structure, γ0 = −10.54 and b = 852 for poly(1-pentene), γ0 = −4.52 and b = 616 for poly(α-olefin).

Correlations between scaling exponent γ and isochoric fragility parameter m V when varying individual molecular parameters. The lines are linear fits according to Eq. (6) with the fitting parameters, γ0 = −8.65 and b = 847 for chains with the poly(propylene) structure, γ0 = −10.54 and b = 852 for poly(1-pentene), γ0 = −4.52 and b = 616 for poly(α-olefin).

## Tables

Cell volume parameters a cell , cohesive energies ε, calculated glass transition temperatures , and experimental glass transition temperatures 56 for various pressures. Using the pressure dependent a cell and ε along with the bending energy E b /k B = 409 K produces the calculated pressure dependence of the glass transition temperature in good agreement with experimental one.

Cell volume parameters a cell , cohesive energies ε, calculated glass transition temperatures , and experimental glass transition temperatures 56 for various pressures. Using the pressure dependent a cell and ε along with the bending energy E b /k B = 409 K produces the calculated pressure dependence of the glass transition temperature in good agreement with experimental one.

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